Basic Variables of Quantum Mechanics for Electrons in Electrostatic and Magnetostatic Fields

We consider a system of $N$ electrons in an external electrostatic {\boldmath $\cal{E}$ } = - {\boldmath $\nabla$ } $v ({\bf{r}})$ and magnetostatic ${\bf{B}} ({\bf{r}}) = \nabla \times {\bf{A}} ({\bf{r}})$ fields, and the Hamiltonian to include the interaction of the latter with both the orbital and spin angular momentum. We prove the one-to-one relationship $\{ v ({\bf{r}}), {\bf{A}} ({\bf{r}}) \} \leftrightarrow \{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$, where $\rho ({\bf{r}})$ and ${\bf{j}} ({\bf{r}})$ are the nondegenerate ground state density and physical current density. The proof accounts for the many-to-one relationship between the $\{ v ({\bf{r}}), {\bf{A}} ({\bf{r}}) \}$ and the ground state $\Psi$. In parallel with the Hohenberg-Kohn theorem proof in which the wave function $\Psi$ of the different physical systems considered is constrained\footnote{X.-Y. Pan and V. Sahni, J. Chem. Phys. \textbf{132}, 164116 (2010)} to a fixed electron number $N$, the corresponding $\Psi$ in our proof is constrained to having the same total orbital ${\bf{L}}$ and spin ${\bf{S}}$ angular momentum. Thus, $\{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$ constitute the basic variables in the rigorous HK sense.